What is the general solution for sin(x+20)=cos(2x+25) ?

The general solution for sin(x+20)=cos(2x+25) is x=(1440n+180)/(12),x=-(3420+12960n)/(36)

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Popular Trigonometry >

sin(x+20)=cos(2x+25)

Solution

sin (x + 2 0^{\circ}) = cos (2 x + 2 5^{\circ})

Solution

x = \frac{144 0 ^{\circ} n + 18 0 ^{\circ}}{12}, x = - \frac{342 0 ^{\circ} + 1296 0 ^{\circ} n}{36}

Radians

x = \frac{π}{12} + \frac{8 π}{12} n, x = - \frac{19 π}{36} - \frac{72 π}{36} n

Solution steps

sin (x + 2 0^{\circ}) = cos (2 x + 2 5^{\circ})

Rewrite using trig identities

sin (x + 2 0^{\circ}) = cos (2 x + 2 5^{\circ})

Use the following identity:

cos (x) = sin (9 0^{\circ} - x)

sin (x + 2 0^{\circ}) = sin (9 0^{\circ} - (2 x + 2 5^{\circ}))

sin (x + 2 0^{\circ}) = sin (9 0^{\circ} - (2 x + 2 5^{\circ}))

Apply trig inverse properties

sin (x + 2 0^{\circ}) = sin (9 0^{\circ} - (2 x + 2 5^{\circ}))

sin (x) = sin (y) \Rightarrow x = y + 2 π n, x = π - y + 2 π n

x + 2 0^{\circ} = 9 0^{\circ} - (2 x + 2 5^{\circ}) + 36 0^{\circ} n, x + 2 0^{\circ} = 18 0^{\circ} - (9 0^{\circ} - (2 x + 2 5^{\circ})) + 36 0^{\circ} n

x + 2 0^{\circ} = 9 0^{\circ} - (2 x + 2 5^{\circ}) + 36 0^{\circ} n, x + 2 0^{\circ} = 18 0^{\circ} - (9 0^{\circ} - (2 x + 2 5^{\circ})) + 36 0^{\circ} n

x + 2 0^{\circ} = 9 0^{\circ} - (2 x + 2 5^{\circ}) + 36 0^{\circ} n : x = \frac{144 0 ^{\circ} n + 18 0 ^{\circ}}{12}

x + 2 0^{\circ} = 9 0^{\circ} - (2 x + 2 5^{\circ}) + 36 0^{\circ} n

Expand

9 0^{\circ} - (2 x + 2 5^{\circ}) + 36 0^{\circ} n : - 2 x + 36 0^{\circ} n + 6 5^{\circ}

9 0^{\circ} - (2 x + 2 5^{\circ}) + 36 0^{\circ} n

- (2 x + 2 5^{\circ}) : - 2 x - 2 5^{\circ}

- (2 x + 2 5^{\circ})

Distribute parentheses

= - (2 x) - (2 5^{\circ})

Apply minus-plus rules

+ (- a) = - a

= - 2 x - 2 5^{\circ}

= 9 0^{\circ} - 2 x - 2 5^{\circ} + 36 0^{\circ} n

Simplify

9 0^{\circ} - 2 x - 2 5^{\circ} + 36 0^{\circ} n : - 2 x + 36 0^{\circ} n + 6 5^{\circ}

9 0^{\circ} - 2 x - 2 5^{\circ} + 36 0^{\circ} n

Group like terms

= - 2 x + 36 0^{\circ} n + 9 0^{\circ} - 2 5^{\circ}

Least Common Multiplier of

2, 36 : 36

2, 36

Least Common Multiplier (LCM)

Prime factorization of

2 : 2

2

2

is a prime number, therefore no factorization is possible

= 2

Prime factorization of

36 : 2 \cdot 2 \cdot 3 \cdot 3

36

36

divides by

236 = 18 \cdot 2

= 2 \cdot 18

18

divides by

218 = 9 \cdot 2

= 2 \cdot 2 \cdot 9

9

divides by

39 = 3 \cdot 3

= 2 \cdot 2 \cdot 3 \cdot 3

2, 3

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 2 \cdot 3 \cdot 3

Multiply each factor the greatest number of times it occurs in either

2

36

= 2 \cdot 2 \cdot 3 \cdot 3

Multiply the numbers:

2 \cdot 2 \cdot 3 \cdot 3 = 36

= 36

Adjust Fractions based on the LCM

Multiply each numerator by the same amount needed to multiply its

corresponding denominator to turn it into the LCM

36

For

9 0^{\circ} :

multiply the denominator and numerator by

18

9 0^{\circ} = \frac{18 0 ^{\circ} 18}{2 \cdot 18} = 9 0^{\circ}

= 9 0^{\circ} - 2 5^{\circ}

Since the denominators are equal, combine the fractions:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{18 0 ^{\circ} 18 - 90 0 ^{\circ}}{36}

Add similar elements:

324 0^{\circ} - 90 0^{\circ} = 234 0^{\circ}

= - 2 x + 36 0^{\circ} n + 6 5^{\circ}

= - 2 x + 36 0^{\circ} n + 6 5^{\circ}

x + 2 0^{\circ} = - 2 x + 36 0^{\circ} n + 6 5^{\circ}

Move

2 0^{\circ}

to the right side

x + 2 0^{\circ} = - 2 x + 36 0^{\circ} n + 6 5^{\circ}

Subtract

2 0^{\circ}

from both sides

x + 2 0^{\circ} - 2 0^{\circ} = - 2 x + 36 0^{\circ} n + 6 5^{\circ} - 2 0^{\circ}

Simplify

x + 2 0^{\circ} - 2 0^{\circ} = - 2 x + 36 0^{\circ} n + 6 5^{\circ} - 2 0^{\circ}

Simplify

x + 2 0^{\circ} - 2 0^{\circ} : x

x + 2 0^{\circ} - 2 0^{\circ}

Add similar elements:

2 0^{\circ} - 2 0^{\circ} = 0

= x

Simplify

- 2 x + 36 0^{\circ} n + 6 5^{\circ} - 2 0^{\circ} : - 2 x + 36 0^{\circ} n + 4 5^{\circ}

- 2 x + 36 0^{\circ} n + 6 5^{\circ} - 2 0^{\circ}

Least Common Multiplier of

36, 9 : 36

36, 9

Least Common Multiplier (LCM)

Prime factorization of

36 : 2 \cdot 2 \cdot 3 \cdot 3

36

36

divides by

236 = 18 \cdot 2

= 2 \cdot 18

18

divides by

218 = 9 \cdot 2

= 2 \cdot 2 \cdot 9

9

divides by

39 = 3 \cdot 3

= 2 \cdot 2 \cdot 3 \cdot 3

2, 3

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 2 \cdot 3 \cdot 3

Prime factorization of

9 : 3 \cdot 3

9

9

divides by

39 = 3 \cdot 3

= 3 \cdot 3

Multiply each factor the greatest number of times it occurs in either

36

9

= 2 \cdot 2 \cdot 3 \cdot 3

Multiply the numbers:

2 \cdot 2 \cdot 3 \cdot 3 = 36

= 36

Adjust Fractions based on the LCM

Multiply each numerator by the same amount needed to multiply its

corresponding denominator to turn it into the LCM

36

For

2 0^{\circ} :

multiply the denominator and numerator by

4

2 0^{\circ} = \frac{18 0 ^{\circ} 4}{9 \cdot 4} = 2 0^{\circ}

= 6 5^{\circ} - 2 0^{\circ}

Since the denominators are equal, combine the fractions:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{234 0 ^{\circ} - 18 0 ^{\circ} 4}{36}

Add similar elements:

234 0^{\circ} - 72 0^{\circ} = 162 0^{\circ}

= 4 5^{\circ}

Cancel the common factor:

9

= - 2 x + 36 0^{\circ} n + 4 5^{\circ}

x = - 2 x + 36 0^{\circ} n + 4 5^{\circ}

x = - 2 x + 36 0^{\circ} n + 4 5^{\circ}

x = - 2 x + 36 0^{\circ} n + 4 5^{\circ}

Move

2 x

to the left side

x = - 2 x + 36 0^{\circ} n + 4 5^{\circ}

Add

2 x

to both sides

x + 2 x = - 2 x + 36 0^{\circ} n + 4 5^{\circ} + 2 x

Simplify

3 x = 36 0^{\circ} n + 4 5^{\circ}

3 x = 36 0^{\circ} n + 4 5^{\circ}

Divide both sides by

3

3 x = 36 0^{\circ} n + 4 5^{\circ}

Divide both sides by

3

\frac{3 x}{3} = \frac{36 0 ^{\circ} n}{3} + \frac{4 5 ^{\circ}}{3}

Simplify

\frac{3 x}{3} = \frac{36 0 ^{\circ} n}{3} + \frac{4 5 ^{\circ}}{3}

Simplify

\frac{3 x}{3} : x

\frac{3 x}{3}

Divide the numbers:

\frac{3}{3} = 1

= x

Simplify

\frac{36 0 ^{\circ} n}{3} + \frac{4 5 ^{\circ}}{3} : \frac{144 0 ^{\circ} n + 18 0 ^{\circ}}{12}

\frac{36 0 ^{\circ} n}{3} + \frac{4 5 ^{\circ}}{3}

Apply rule

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{36 0 ^{\circ} n + 4 5 ^{\circ}}{3}

Join

36 0^{\circ} n + 4 5^{\circ} : \frac{144 0 ^{\circ} n + 18 0 ^{\circ}}{4}

36 0^{\circ} n + 4 5^{\circ}

Convert element to fraction:

36 0^{\circ} n = \frac{36 0 ^{\circ} n 4}{4}

= \frac{36 0 ^{\circ} n \cdot 4}{4} + 4 5^{\circ}

Since the denominators are equal, combine the fractions:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{36 0 ^{\circ} n \cdot 4 + 18 0 ^{\circ}}{4}

Multiply the numbers:

2 \cdot 4 = 8

= \frac{144 0 ^{\circ} n + 18 0 ^{\circ}}{4}

= \frac{\frac{144 0 ^{\circ} n + 18 0 ^{\circ}}{4}}{3}

Apply the fraction rule:

\frac{\frac{b}{c}}{a} = \frac{b}{c \cdot a}

= \frac{144 0 ^{\circ} n + 18 0 ^{\circ}}{4 \cdot 3}

Multiply the numbers:

4 \cdot 3 = 12

= \frac{144 0 ^{\circ} n + 18 0 ^{\circ}}{12}

x = \frac{144 0 ^{\circ} n + 18 0 ^{\circ}}{12}

x = \frac{144 0 ^{\circ} n + 18 0 ^{\circ}}{12}

x = \frac{144 0 ^{\circ} n + 18 0 ^{\circ}}{12}

x + 2 0^{\circ} = 18 0^{\circ} - (9 0^{\circ} - (2 x + 2 5^{\circ})) + 36 0^{\circ} n : x = - \frac{342 0 ^{\circ} + 1296 0 ^{\circ} n}{36}

x + 2 0^{\circ} = 18 0^{\circ} - (9 0^{\circ} - (2 x + 2 5^{\circ})) + 36 0^{\circ} n

Expand

18 0^{\circ} - (9 0^{\circ} - (2 x + 2 5^{\circ})) + 36 0^{\circ} n : 18 0^{\circ} + 2 x - 6 5^{\circ} + 36 0^{\circ} n

18 0^{\circ} - (9 0^{\circ} - (2 x + 2 5^{\circ})) + 36 0^{\circ} n

Expand

9 0^{\circ} - (2 x + 2 5^{\circ}) : - 2 x + 6 5^{\circ}

9 0^{\circ} - (2 x + 2 5^{\circ})

- (2 x + 2 5^{\circ}) : - 2 x - 2 5^{\circ}

- (2 x + 2 5^{\circ})

Distribute parentheses

= - (2 x) - (2 5^{\circ})

Apply minus-plus rules

+ (- a) = - a

= - 2 x - 2 5^{\circ}

= 9 0^{\circ} - 2 x - 2 5^{\circ}

Simplify

9 0^{\circ} - 2 x - 2 5^{\circ} : - 2 x + 6 5^{\circ}

9 0^{\circ} - 2 x - 2 5^{\circ}

Group like terms

= - 2 x + 9 0^{\circ} - 2 5^{\circ}

Least Common Multiplier of

2, 36 : 36

2, 36

Least Common Multiplier (LCM)

Prime factorization of

2 : 2

2

2

is a prime number, therefore no factorization is possible

= 2

Prime factorization of

36 : 2 \cdot 2 \cdot 3 \cdot 3

36

36

divides by

236 = 18 \cdot 2

= 2 \cdot 18

18

divides by

218 = 9 \cdot 2

= 2 \cdot 2 \cdot 9

9

divides by

39 = 3 \cdot 3

= 2 \cdot 2 \cdot 3 \cdot 3

2, 3

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 2 \cdot 3 \cdot 3

Multiply each factor the greatest number of times it occurs in either

2

36

= 2 \cdot 2 \cdot 3 \cdot 3

Multiply the numbers:

2 \cdot 2 \cdot 3 \cdot 3 = 36

= 36

Adjust Fractions based on the LCM

Multiply each numerator by the same amount needed to multiply its

corresponding denominator to turn it into the LCM

36

For

9 0^{\circ} :

multiply the denominator and numerator by

18

9 0^{\circ} = \frac{18 0 ^{\circ} 18}{2 \cdot 18} = 9 0^{\circ}

= 9 0^{\circ} - 2 5^{\circ}

Since the denominators are equal, combine the fractions:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{18 0 ^{\circ} 18 - 90 0 ^{\circ}}{36}

Add similar elements:

324 0^{\circ} - 90 0^{\circ} = 234 0^{\circ}

= - 2 x + 6 5^{\circ}

= - 2 x + 6 5^{\circ}

= 18 0^{\circ} - (- 2 x + 6 5^{\circ}) + 36 0^{\circ} n

- (- 2 x + 6 5^{\circ}) : 2 x - 6 5^{\circ}

- (- 2 x + 6 5^{\circ})

Distribute parentheses

= - (- 2 x) - (6 5^{\circ})

Apply minus-plus rules

- (- a) = a, - (a) = - a

= 2 x - 6 5^{\circ}

= 18 0^{\circ} + 2 x - 6 5^{\circ} + 36 0^{\circ} n

x + 2 0^{\circ} = 18 0^{\circ} + 2 x - 6 5^{\circ} + 36 0^{\circ} n

Move

2 0^{\circ}

to the right side

x + 2 0^{\circ} = 18 0^{\circ} + 2 x - 6 5^{\circ} + 36 0^{\circ} n

Subtract

2 0^{\circ}

from both sides

x + 2 0^{\circ} - 2 0^{\circ} = 18 0^{\circ} + 2 x - 6 5^{\circ} + 36 0^{\circ} n - 2 0^{\circ}

Simplify

x + 2 0^{\circ} - 2 0^{\circ} = 18 0^{\circ} + 2 x - 6 5^{\circ} + 36 0^{\circ} n - 2 0^{\circ}

Simplify

x + 2 0^{\circ} - 2 0^{\circ} : x

x + 2 0^{\circ} - 2 0^{\circ}

Add similar elements:

2 0^{\circ} - 2 0^{\circ} = 0

= x

Simplify

18 0^{\circ} + 2 x - 6 5^{\circ} + 36 0^{\circ} n - 2 0^{\circ} : 2 x + 18 0^{\circ} + 36 0^{\circ} n - 8 5^{\circ}

18 0^{\circ} + 2 x - 6 5^{\circ} + 36 0^{\circ} n - 2 0^{\circ}

Group like terms

= 2 x + 18 0^{\circ} + 36 0^{\circ} n - 2 0^{\circ} - 6 5^{\circ}

Least Common Multiplier of

9, 36 : 36

9, 36

Least Common Multiplier (LCM)

Prime factorization of

9 : 3 \cdot 3

9

9

divides by

39 = 3 \cdot 3

= 3 \cdot 3

Prime factorization of

36 : 2 \cdot 2 \cdot 3 \cdot 3

36

36

divides by

236 = 18 \cdot 2

= 2 \cdot 18

18

divides by

218 = 9 \cdot 2

= 2 \cdot 2 \cdot 9

9

divides by

39 = 3 \cdot 3

= 2 \cdot 2 \cdot 3 \cdot 3

2, 3

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 2 \cdot 3 \cdot 3

Multiply each factor the greatest number of times it occurs in either

9

36

= 3 \cdot 3 \cdot 2 \cdot 2

Multiply the numbers:

3 \cdot 3 \cdot 2 \cdot 2 = 36

= 36

Adjust Fractions based on the LCM

Multiply each numerator by the same amount needed to multiply its

corresponding denominator to turn it into the LCM

36

For

2 0^{\circ} :

multiply the denominator and numerator by

4

2 0^{\circ} = \frac{18 0 ^{\circ} 4}{9 \cdot 4} = 2 0^{\circ}

= - 2 0^{\circ} - 6 5^{\circ}

Since the denominators are equal, combine the fractions:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{- 18 0 ^{\circ} 4 - 234 0 ^{\circ}}{36}

Add similar elements:

- 72 0^{\circ} - 234 0^{\circ} = - 306 0^{\circ}

= \frac{- 306 0 ^{\circ}}{36}

Apply the fraction rule:

\frac{- a}{b} = - \frac{a}{b}

= 2 x + 18 0^{\circ} + 36 0^{\circ} n - 8 5^{\circ}

x = 2 x + 18 0^{\circ} + 36 0^{\circ} n - 8 5^{\circ}

x = 2 x + 18 0^{\circ} + 36 0^{\circ} n - 8 5^{\circ}

x = 2 x + 18 0^{\circ} + 36 0^{\circ} n - 8 5^{\circ}

Move

2 x

to the left side

x = 2 x + 18 0^{\circ} + 36 0^{\circ} n - 8 5^{\circ}

Subtract

2 x

from both sides

x - 2 x = 2 x + 18 0^{\circ} + 36 0^{\circ} n - 8 5^{\circ} - 2 x

Simplify

- x = 18 0^{\circ} + 36 0^{\circ} n - 8 5^{\circ}

- x = 18 0^{\circ} + 36 0^{\circ} n - 8 5^{\circ}

Divide both sides by

- 1

- x = 18 0^{\circ} + 36 0^{\circ} n - 8 5^{\circ}

Divide both sides by

- 1

\frac{- x}{- 1} = \frac{18 0 ^{\circ}}{- 1} + \frac{36 0 ^{\circ} n}{- 1} - \frac{8 5 ^{\circ}}{- 1}

Simplify

\frac{- x}{- 1} = \frac{18 0 ^{\circ}}{- 1} + \frac{36 0 ^{\circ} n}{- 1} - \frac{8 5 ^{\circ}}{- 1}

Simplify

\frac{- x}{- 1} : x

\frac{- x}{- 1}

Apply the fraction rule:

\frac{- a}{- b} = \frac{a}{b}

= \frac{x}{1}

Apply rule

\frac{a}{1} = a

= x

Simplify

\frac{18 0 ^{\circ}}{- 1} + \frac{36 0 ^{\circ} n}{- 1} - \frac{8 5 ^{\circ}}{- 1} : - \frac{342 0 ^{\circ} + 1296 0 ^{\circ} n}{36}

\frac{18 0 ^{\circ}}{- 1} + \frac{36 0 ^{\circ} n}{- 1} - \frac{8 5 ^{\circ}}{- 1}

Apply rule

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{18 0 ^{\circ} + 36 0 ^{\circ} n - 8 5 ^{\circ}}{- 1}

Apply the fraction rule:

\frac{a}{- b} = - \frac{a}{b}

= - \frac{18 0 ^{\circ} + 36 0 ^{\circ} n - 8 5 ^{\circ}}{1}

Join

18 0^{\circ} + 36 0^{\circ} n - 8 5^{\circ} : \frac{342 0 ^{\circ} + 1296 0 ^{\circ} n}{36}

18 0^{\circ} + 36 0^{\circ} n - 8 5^{\circ}

Convert element to fraction:

18 0^{\circ} = 18 0^{\circ}, 36 0^{\circ} n = \frac{36 0 ^{\circ} n 36}{36}

= 18 0^{\circ} + \frac{36 0 ^{\circ} n \cdot 36}{36} - 8 5^{\circ}

Since the denominators are equal, combine the fractions:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{18 0 ^{\circ} 36 + 36 0 ^{\circ} n \cdot 36 - 306 0 ^{\circ}}{36}

18 0^{\circ} 36 + 36 0^{\circ} n \cdot 36 - 306 0^{\circ} = 342 0^{\circ} + 1296 0^{\circ} n

18 0^{\circ} 36 + 36 0^{\circ} n \cdot 36 - 306 0^{\circ}

Add similar elements:

648 0^{\circ} - 306 0^{\circ} = 342 0^{\circ}

= 342 0^{\circ} + 2 \cdot 648 0^{\circ} n

Multiply the numbers:

2 \cdot 36 = 72

= 342 0^{\circ} + 1296 0^{\circ} n

= \frac{342 0 ^{\circ} + 1296 0 ^{\circ} n}{36}

= - \frac{\frac{342 0 ^{\circ} + 1296 0 ^{\circ} n}{36}}{1}

Apply the fraction rule:

\frac{a}{1} = a

= - \frac{342 0 ^{\circ} + 1296 0 ^{\circ} n}{36}

x = - \frac{342 0 ^{\circ} + 1296 0 ^{\circ} n}{36}

x = - \frac{342 0 ^{\circ} + 1296 0 ^{\circ} n}{36}

x = - \frac{342 0 ^{\circ} + 1296 0 ^{\circ} n}{36}

x = \frac{144 0 ^{\circ} n + 18 0 ^{\circ}}{12}, x = - \frac{342 0 ^{\circ} + 1296 0 ^{\circ} n}{36}

x = \frac{144 0 ^{\circ} n + 18 0 ^{\circ}}{12}, x = - \frac{342 0 ^{\circ} + 1296 0 ^{\circ} n}{36}

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Frequently Asked Questions (FAQ)

What is the general solution for sin(x+20)=cos(2x+25) ?
The general solution for sin(x+20)=cos(2x+25) is x=(1440n+180)/(12),x=-(3420+12960n)/(36)